Homology of tropical fans
arXiv:2105.01504
Abstract
The aim of this paper is to study homological properties of tropical fans and to propose a notion of smoothness in tropical geometry, which goes beyond matroids and their Bergman fans and which leads to an enrichment of the category of smooth tropical varieties. Among the resulting applications, we prove the Hodge isomorphism theorem which asserts that the Chow rings of smooth unimodular tropical fans are isomorphic to the tropical cohomology rings of their corresponding canonical compactifications, and prove a slightly weaker statement for any unimodular fan. We furthermore introduce a notion of shellability for tropical fans and show that shellable tropical fans are smooth and thus enjoy all the nice homological properties of smooth tropical fans. Several other interesting properties for tropical fans are shown to be shellable. Finally, we obtain a generalization, both in the tropical and in the classical setting, of the pioneering work of Feichtner-Yuzvinsky and De Concini-Procesi on the cohomology ring of wonderful compactifications of complements of hyperplane arrangements. The results in this paper form the basis for our subsequent works on Hodge theory for tropical and non-Archimedean varieties.
arXiv comment85 pages, 1 figure
References in corpus (19)
- The tropicalization of the moduli space of curves
- Tropical fans and the moduli spaces of tropical curves
- Moduli spaces of rational tropical curves
- Formes différentielles réelles et courants sur les espaces de Berkovich
- A tropical approach to non-archimedean Arakelov theory
- Combinatorial Lefschetz theorems beyond positivity
- Lefschetz (1,1)-theorem in tropical geometry
- Orlik-Solomon-type presentations for the cohomology algebra of toric arrangements
- The Orlik-Solomon Algebra and the Bergman Fan of a Matroid
- Local systems on complements of arrangements of smooth, complex algebraic hypersurfaces
- A semi-small decomposition of the Chow ring of a matroid
- Hodge theory for tropical varieties
- Lefschetz section theorems for tropical hypersurfaces
- Filtered geometric lattices and Lefschetz Section Theorems over the tropical semiring
- Complex Tropical Currents, Extremality, and Approximations
- Tropical Clemens-Schmid sequence and existence of tropical cycles with a given cohomology class
- A guide to tropical modifications
- Tropical analogs of Milnor -groups and tropicalizations of Zariski-Riemann spaces
- On tropical cohomology of smooth algebraic varieties